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Next: Green's theorem in the Up: Vector Fields, Work, Circulation Previous: Work of a force.

Flow, circulation and flux.

The curl of a plane vector field has been already defined in section  5.

If the vector field $\overrightarrow{F} $ represent the velocity field of a fluid, the integral in  8.11 defines the flow of the vector field along the curve $\mathcal{C}$.

Definition 8.14   Let $\mathcal{C}$ be a smooth curve given by the parametrization $\overrightarrow{r} (t)= g_1(t) \overrightarrow{i} + g_2(t) \overrightarrow{j} + g_3(t) \overrightarrow{k} $, where $a \leq t \leq b$. We denote by $\overrightarrow{F} = M(x,y,z) \overrightarrow{i} + N(x,y,z) \overrightarrow{j} +P(x,y,z) \overrightarrow{k} $ a continuous velocity field. The flow of $\overrightarrow{F} $ along $\mathcal{C}$ is equal to

\begin{displaymath}\int_a^b \overrightarrow{F}\cdot \overrightarrow{T}\; ds
\end{displaymath}

This integral is called a flow integral.

If $\mathcal{C}$ is closed loop, i.e. $\overrightarrow{r} (a)= \overrightarrow{r} (b)$, then the flow is called the circulation of $\overrightarrow{F} $ around $\mathcal{C}$.

Remark 8.15   This definition is valid for a vector field and a curve in the plane.

Example 8.16 (In the plane.)   Take $\overrightarrow{F} =y \overrightarrow{i} + x \overrightarrow{j} $ and let $\mathcal{C}$ be the curve with parametrization $\overrightarrow{r} (t)= \cos t \overrightarrow{i} + \sin t \overrightarrow{j} $, where $0 \leq t \leq 2 \pi$.
  
Figure 4: A loop and a tangent vector.
\begin{figure}
\mbox{\epsfig{file=TrigoCircle.eps,height=4cm} }
\end{figure}

Example 8.17 (In the space.)   Take $\overrightarrow{F} =y \overrightarrow{i} + x \overrightarrow{j} + z \overrightarrow{k} $ and let $\mathcal{C}$ be the curve with parametrization $\overrightarrow{r} (t)= \cos t \overrightarrow{i} + \sin t \overrightarrow{j} +t \overrightarrow{k} $, where $0 \leq t \leq 2 \pi$ (a loop of an helix, as displayed on Fig.  1).

Definition 8.18   Let $\mathcal{C}$ be a smooth curve in the plane. At every point of $\mathcal{C}$ is defined a unit normal vector $\overrightarrow{n} $ pointing outwards. Let now $\overrightarrow{F} = M(x,y) \overrightarrow{i} + N(x,y) \overrightarrow{j} $ be a vector field in the plane defined over a domain containing the curve $\mathcal{C}$.

The flux of $\overrightarrow{F} $ across the curve $\mathcal{C}$ is the line integral

\begin{displaymath}\int_{\mathcal{C}} \overrightarrow{F}\cdot \overrightarrow{n}\; ds.
\end{displaymath}

% latex2html id marker 10718
\fbox{
\begin{minipage}{10cm}
\begin{center}
\under...
... = \; \oint_{\mathcal{C}} M \; dy \; - \; N \;dx
\end{equation*}\end{minipage} }

Example 8.19   Take $\overrightarrow{F} =y \overrightarrow{i} + x \overrightarrow{j} $ and let $\mathcal{C}$ be the curve with parametrization $\overrightarrow{r} (t)= \cos t \overrightarrow{i} + \sin t \overrightarrow{j} $, where $0 \leq t \leq 2 \pi$.

On the curve $\mathcal{C}$:

Therefore:

\begin{displaymath}M \; dy \; - \; N \;dx = \sin t \cos t \; dt - \cos t ( - \sin t ) \; dt = \sin 2t \; dt.
\end{displaymath}

And we have:

\begin{displaymath}\text{Flux of $F$\space across } \mathcal{C} \; = \;
\int_0^{2 \pi} \sin 2t \; dt = 0.
\end{displaymath}


next up previous contents
Next: Green's theorem in the Up: Vector Fields, Work, Circulation Previous: Work of a force.
Noah Dana-Picard
2001-05-30